270 Mr. Moseley on Caustics. 



sin QSN sin PSM 



= 7rt = 



sin PS Q sinNSM 



sin (PSQ -(- PSN) _ sin (PSN + NSM) 

 sin PSQ ~ sinNSM 



.'. COS PSN + sin PSN cot PSQ = cos PSN + sin PSN 



cot NSM 



.-. cot PSQ = cot NSM 



PSQ = NSM 



.-. QSN = PSM, 



.: ^ = S- (1) 



Also, R3= r^ + 4 - — cos QPS 



TO- m 



Cos QPS = cos { < inc. — < refr.} 

 Cos < inc. = — 



.'. mi^ COS QPS = ±p \/' {m^-l)r' +p^ - 7^ + / 



.-. R« = /^.(l + ^)-^,{±p V{nr-iy+p^ + T^-p^ \ 



... ,^^ (Ay= K + l)(^y-2 \ ±yK-l)(p^+l - 



... .. (^y = (.^ + 1) (4y- 2 \ ± N/K^=^i)(iy7i 

 -ay+M ; (^) 



Eliminating (r) and {p) between the equations (1) and (2), 

 and the given equation p =J'r of the refracting curve we shall 

 obtain the required equation in R and P. 



Thus in the logarithmic spiral, since — = a, we have by 

 equation (1) -rr = ^> •'• the involute to the caustic is also a 



logarithmic spiral similar to the refracting curve. Also the 

 evolute is in this case similar to its involute; therefore 



the 



