182 Mr. T. S. Davies on Bernoulli's Solution 



In this expression q is the variable, and the differentiation 

 being performed with respect to it, we have, after transposition, 



(COS X — COS q f COS q) COSeC q 



S/ (Sin 9 X — COS - p) + 2 COS q f COS A COS q — COS 2 q 

 (COS X — COS q u COS q) COSeC £ 



(3) 



\/sin 3 X — cos 2 ^ ;/ + 2 cos ^cos Xcosg — cos 2 g 



Squaring these to take the radical from the denominator, and 

 transposing all to one side, it takes the form 



(COS X — COS q / COS q)* 

 Sin - X — COS 3 q -f- 2 COS X COS q / COS q — COS 2 q 

 (COS X -— COS q u COS qf 



r\ ^ = cosec * p % ~s ■ — — 



5 t Sill-X— COS 3 q + 2 COS X COS q / COS q — COS 2 



(cos X — nos />.. rns nS 2 \ 



(4) 



sin 2 X — cos 2 g w + 2 cos X cos q u cos g — cos g 2 J 

 This expression is probably not much unlike that which 

 John Bernoulli obtained, allowance being made for the greater 

 generality of its objects and the difference of the notations. 

 As, however, he merely describes it, we can only conjec- 

 ture what it might have been*: at all events it does not 

 greatly differ from that given by D'Alembert, in the Encyclo- 

 pedic Methodiquc, Art. CitfcpuscuLE. Let us now attend to 

 the reduction of the factor within the brackets : and first, cross 

 multiply, and arrange the resulting terms according to the 

 powers of cos q ; then divide all the terms by cos 3 g tl — cos 2 q t : 

 our result is 



cos*e — 2cosX M ^'cos 3 p — sin 2 X cos 2 p 



5 COS q / + COS q u 5 5 



1 -f COS p. COS p.. 



+ 2 cos X — & ^ cos p — cos 2 X ; 



COS q l + COS q n s 



which may again be readily changed into 



(l-sin 2 ? ) 2 + 2cosX. 1 -±^ S -f^^^. cos* (1 -cos 2 *) 

 v 5/ cos^+ cosq tl 5 v 5/ 



— sin 2 X(l— sin 2 g) — cos 2 X ; 



and this is at once converted into 



sin 



4cos 3 P-2cosX 1 " fCOSg/COSg// cosg + cos 2 X? (5) 



5 £ 5 cos^ + cosg,, 5 3 v ' 



Inserting instead of the bracketed factor of (4) its value 

 given by (5), we have that equation (4) at once transformed 

 into 



cos 2 p-2cosX 1 -±^ S ^^ C — ^cos* + cos 2 X = (6) 



5 COS q / + COS §// 



* Johan. Bernoulli, Opera Ovinia, tome i. p. G4. 



