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A. C. L. Wilkinson

Abstract

Gentlemen,

We are met together to-day for the first time to fulfil one of the most essential functions of a Society such as oars, a function sucoinotly stated in our First Constitution in paragraph 7-the promoting of the mutual knowledge and intercourse of members and their co-operation in matters of common or general interest with the objects of the Society- For myself, I consider it a great privilege to realise, that I have now become personally acquainted with many known to me previously by name or reputation only, who have rendered so many services to that object for which our society was founded-the promotion of mathematical study and research in India.

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A. C. L. Wilkinson

Abstract

ABC is a triangle; A'B'C the middle points of the sides forming the medial triangle; DEF the orthocentric or pedal triangle; S the circumcentre; 0 the orthocentre; A₁, B₁, C₁ the middle points of OA, OB, OC; G the centroid; I the in-centre; N the nine point circle centre; K the symmedian point; a,b,c the points of contact of the inscribed circle with the sides; π the Feuerbach point for the incircle; d the point of contact with the Steiner ellipse of the common tangent of the Steiner ellipse, inscribed and nine point circles; T the Gergonne point of intersection of Aa, Bb, Cc.

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A. C. L. Wilkinson

Abstract

If α,β,γ are the areal coordinates of a point, α+β+γ= 1. Also (α,β,γ) is the centroid of masses α,β,γ placed at A,B,C the

vertice 3 of the triangle of reference.

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A. C. L. Wilkinson

Abstract

I propose to solve analytically the problem of determining all tetrahedra for which the three shortest distances between pair.s of opposite edges intersect in a point; while, however, this is the direct object of the following sections a number of other results that arise out of the work will be noticed.

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A. C. L. Wilkinson

Abstract

The radius of curvature of a curve referred to rectangular axes is given by 

1/P=d²y/ds² dx/ds - d²y/ds,dy/ds.

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A. C. L. Wilkinson

Abstract

if f(αβγδ)≡(a,b,c,d,f,g,h,u,v,w)(αβγδ)2=0 represent a sphere anrl (αβγδ) is its centra, then by eonsideiing its intersection.

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A. C. L. Wilkinson

Abstract

The discrimination of the conics given by the general equation.

Let uα²+vβ²+wγ²+2u^'βγ+2v^'γα+2w^'αβ=0

be the conic, which will be denoted by

(u,v,w,u^',v^',w^')(α,β,γ)²=0.

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A. C. L. Wilkinson

Abstract

A fundamental theorem of the general bicircular quantic, given by Casey, is the following:-

A bicircular quartic can be generated in four and only four ways as the envelope of a circle which cuts a fixed circle orthogonally and whose centre moves on a fixed central conic. This fixed conic is called a focal conic and the four focal conies corresponding to the four methods of generating the quartic are confocal, their foci being the double foci of the quartic.

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A. C. L. Wilkinson

Abstract

A bicircular quartic can be generated in four and only four ways as the envelope of a circle which cuts a fixed circle orthogonally and whose centre moves on a fixed central conic. This fixed conic is called a focal conic and the four focal conies corresponding to the four methods of generating the quartic are confocal, their foci being the double foci of the quartic.

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A. C. L. Wilkinson

Abstract

In the tig. on page 50 where the circle of inversion cuts the focal ellipse ;n four real points, there are four real common tangents to the ellipse and circle αα, ββ, γγ, δδ.

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A. C. L. Wilkinson

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Prove that ∫ Φ(a+ix)-Φ(a-ix)/2ix dx=π/2{Φ(∞)-Φ(a)}.

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A. C. L. Wilkinson

Abstract

1. Consider a pair of rectangular axes rotating uniformly about the origin.

Let (u, v) be the co-ordinates of a point referred to the moving axes at any time, and let (X,Y) be the co-ordinates of the same point referred to the initial position of the axes, and let 0 be the angle turned through from the initial position.

Then

X=i cosθ -v sin θ.

Y= u sinθ+v cos θ.

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A. C. L. Wilkinson

Abstract

It has been suggested to me that some articles dealing with mathematical subjects somewhat in advance of the standard English text books will be welcomed by those readers of the Indian Mathematical Journal who have not access to the great modern French and German treatises on Pure Mathematics.

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A. C. L. Wilkinson

Abstract

Consider a curve PQR… ; along the tangents at P,Q,R…take lengths PP', QQ', RR'…equal to the arcs PB, QB, RB…, then the curve P'Q'R' . . . Is an inrolute of PQR... By varying the position of B, we obtain an infinite number of involutes.

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A. C. L. Wilkinson

Abstract

The usual discussion of the rules of convergency and divergency of series given in elementry treatises is generally far from satisfactory for two reasons : 1° it is generally assumed that there is a definite limiting value of the ratio u_u+1 : as n tends to infinity; 2° the process of proceeding to the limit for the convergency tests is far from rigorous.

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A. C. L. Wilkinson

Abstract

Consider the families of surfaces

f₁(x,y,z) = λ, f₂(x,y,z)=μ,f₁(x,y,z)=v,

x=Φ(λ,μv), y=Φ(λ,μ,v) z=x(&#155,μv).

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A. C. L. Wilkinson

Abstract

We may suppose any surface given by 

x = Φ(u,v), y=Φ(u,v), z=x(u,r)

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A. C. L. Wilkinson

Abstract

Surfaces of which one set of lines of curvature are plane curves lying in parallel planes.

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