Formula for the Hyper geometric Function A 4 (^). 171 

 Hence A 4 (- 5 l + «, 1 + A l + 7< 1 + 5) 



ri+an+)8ri+7ri + 8' 



25/2^3/2 



-i(«+0+r+*+3/2) 



M 



where B is the preceding large bracket, according as </> is 

 positive or negative, the last term being omitted if (f> is zero. 

 Hence in particular, p being real, 



oA 4 (>; l+«, 14/3, 1 + % 1 + 8) 



1 + «.1 + /3.1+ 7 .1 + S •• 



ri+ a ri+/3ri +7 ri+8 , (a+/3+T+0 . +3/2) 



05/2_3/2 r 



2"/-7T 



pp* + 2cos{V-^(a + /3 + 7 + ^ + 3/2) j] 



and 



1_ r l 



1 + a. 14-/3.1 + 7.1 + 8^ 



^ ri+ari+^ri+7ri + 8 / 4p ^ 



25/2^3/2 



(• 



xcosj4/)isinj-j : (a+/3 + 7+8+3/2) ) 



Special forms near infinity. 



Series of the type A 4 occur as solutions of equations of 

 the type 



h{§ + a)(§ + h\§ + c)<o = zw, 



which are of frequent occurrence in problems of lateral 



vibrations of bars whose section is not uniform. 



If we put 



& = z v y, 



the equation becomes 



(^-r)(Ha-r)(H^-r)(Hc-r)?/ = zy. 



The general solution of the original equation — if we formally 

 exclude the cases when logarithmic solutions are involved — 



