332 Prof. Challis on a Mathematical Theory of 



of OA about OZ, and let cot — the angle which the plane AOZ 

 makes with the plane YOZ at the time t. 



The force of gravity being g, the resolved parts in the direc- 

 tions of OX, OY, OZ are 



g . cos AOX, g . cos AOY, g . cos AOZ ; 

 or 



g . sin X sin cot, g . sin "A. cos cot, g . cos X. 



The accelerative force of the tension of the cord being T, the 

 resolved parts in the same directions are 



_ T* _ Tty _ Tz 



a ' a ' a 



Consequently, 



(f-x ■ : Tx 



-To =0 sin A, sin cot 



dr * a 



d *y ■ ^ t T V 



-yq =7sni\ cos oh 



dr " a 



d*z , Tz 



— — = <7C0SX . 



dt* J a 



These are the differential equations it was proposed to find. 



It will be convenient to transform these equations into others 

 containing new rectangular coordinates x', y', z' of P, referred to 

 the same origin O, the axis of x' being in the plane AOZ at right 

 angles to OA, the axis of y' perpendicular to this plane, and the 

 axis of z' coincident with OA. It will be assumed that the po- 

 sitive direction of x' is towards the plane YOX, the positive 

 direction of y' towards the plane YOZ, and the positive direction 

 of z' that of the. action of gravity. This being premised, the 

 following relations between the two systems of coordinates may 

 be readily found : 



x = (z 1 sin X + x' cos X) sin cot — y' cos cot 



y= (z' sin X -f x' cos X) cos cot + y' sin cot 



z = z' cosX — a'sinX. 



By substituting these values of x, y, z in the foregoing equa- 

 tions, the following results may be obtained : 



. — - = 2<y cosX-; — h a) 2 co$\(z' sinX + a/ cosX) 



dr a at 



cPy> T V ' , (&> . . , dx< X -, 



d*z' IV n -ft,/ . , . , 



=q 2(0 sin X-^ — I- or sin X\z sinX + .r cosX). 



dr a <it 



