166 



(IV.) To efFect these double integrations, let us put the functions S, Q, under the form 

 « = 2.S^.,„-.^"".3"", Q=2.Q„,„^y'»S'»', (Um), 



and let us change the differential product dx'. dy' to dx'. di, which is permitted, because in 

 forming this product y varies independently of x' . In this manner the expressions (RP)) 

 become 



Jldy/^JSdx', f^dV^fQ.dx', 

 in which 



/^Sdx' ^ 2. Sm,m' 



m + 1 



^ V (V(7)) 



fQdx-=-2.Q„,,„,. -J. L_ . S'« 



»H+1 



j;',, x'„ being the extreme values of x', corresponding to any given value of 3, that is, the 

 abscissae of the points where the little circular circumference is crossed by any given parallel to 

 the section of the caustic surface. To determine these values, we have the equation 



x"- + (/o + ^Y = r^, (WW) 



y'o being the ordinate of the section, and r the radius of the circle : and putting this equation 

 under the form 



x'^ + y^i = r _ 2^'o3, in which 3' = ± ^^/^ — 3^, (XO) ) 



we can, by Laplace's theorem, develope x'^+i according to the positive integer powers of 3, 5', 

 the term of least dimension being 3""'^'; from which it follows that the integrals (V(^)) may be 

 put under the form 



fSdx' = 23' S' = 22.S'„ „/>3«"''+J -) 



I (YW) 



/ Qdx' = 25' Q* = 22. Ctny .3»3'«»'+i 

 in which 



S'ofi = Sofi, Q'ofl = Qo.oi (Z^'') 



Sofi, Qo,o. being the values of S, Q, assigned by the formulae (S(')). Multiplying (Y(')) by 

 2</^3, integrating with reference to 3 from 3 = to 3 = r, and putting for abridgment 



I„y=fl.z''{l-zY-^*(iy'z, (AW), 



we find finally 



SW = 4rl.2.S'„.„,. /„,„,. r»+2«', 



Q(') = 4rJ. 2. Q',^,;. /„,„,. r» -1-5'"; 



} 



and therefore when we suppose r infinitely small. 



