SDIIL.VR TO THOSE OF HYDRODYNAMICS. 367 



?£t^=* i=0,l,2,....«-l, 



, _ >i — 1 9 _ (n — 2) cos oj , (n — 3) cos a, 



V a — ~~^~> V" a l — a.sgina! ' ^ 7 " a2 ~ a, 2 sina, sin 2 a, ' 



[n — (t + 1)] cosa,- , _ „ 



V" a j — ! J = >-1 . > V °n-l — u - 



sin a,- H sin 2 «,- 



Substituting in y 2 ^, replacing a^ by r, and multiplying through by r we have 



^ J» 1 *» 1 *# . . 1 _**_ 



A* - I *. + ^ + sin 2 ^ Atf"*-- ' ' * + >=ff l sin2 <*«.« ■»■•■•■ "^ i--- 2 gin2 «-, 



<M _ rf<£ (»— 2)cos«, d<£ (w-3)cos«o rf<ft [» — (i + l)] costtj 



+ r d7V l ~ d^ sin «, da, sinojsin 2 *, •••* do, ana/Tof rin«^ 



+ .... '" (16) 



or, arranging a little differently, 



rl-(r4>) Id/. d<j> \ 1 d ( . d<f> \ 



BUI a, 3T I + • • • 



[' 



1 d / . </<£ \ . 1 d ( . (ty\ 



+ sln^ ^ ^ SIU «i <7^ J + sin« 2 . S m 2 ai d^ ^ 8in ^ da, J 



+ . ;-■-» . . da\ (, smai 5 J + • • • • + '-jr'.m-« A._J 



sin «, II sin- «, \ ' 



r / „ > d& , . „ . cos «! d<f> 



+ [(»-») ' ? + O^ 3 ) STi, ^ + ^~ 4 ) 



j_r rion oosa> 1 d * ■ i cos "--2 _1 dA ~| _ n7 x 



II sin 2 «,^ sra«„_ 2 n sln a n 



Denote by I 7 the region of all values of x 1} .r 2 .... .r„ for which 



.Tf + x 2 2 + .... + .r„ 2 = i? 2 (IS) 



i. e. E is the superior limit of r. Denote by S the boundary of T; then S signifies 

 the totality of all systems of values for which 



^ + xi + . . . . + av = i? 2 . (19) 



The equation y 2 <£ = holds throughout the entire region T and right up to the 

 boundary S. Then it can be proved just as in the case of three variables that there 

 is one and only one l'eal function <f> of the variables x, that for the whole region T is 

 single valued and continuous, and at the boundary 8 coincides with an arbitrary 

 function spread out continuously over 8. The first and second differential coefficients 



