838 PROCEEDINGS OF THE AMERICAN ACADEMY 



the plane of ay, and passing through the extremities of -|- (3 and — ^; 

 whereas the second set gives no points of the surface contained between 

 these planes. These might be called, therefore, supplemental equations 

 of the surface. 



If the equations be reduced to scalar forms in rectangular co-ordi- 

 nates, they become identical, and their limitations disappear. Thus 

 we have : — 



x^=:z a sin x Shy, x.^^=^ a Sha; Chy, 



y^ = h cos X, y.,= ± b Chx, 



2j = c sin X Chi/, 22 = Shx Shy, 



and hence 



^j_.Vii_|_fr , ■!£ \^ll. \ 5l 



Imaginary values of the variables x and y in the equations would, 

 however, make either set complete. 



26°. Formula (11), for finding volumes, is 



v= i//sc'c>'iC'V 



From the equation of the general ellipsoid, it is at once evident that 



Vijq\ = V«(T, Yaa' = Y^y, 

 whence follows 



^QQ'i'J-i ^^ So'gVoo'j = sin x Sa'Vaa 



= sin x SiiVoa' = sin x S«(3j' =: abc sin x. 



The ellipsoidal volume is therefore 



_- abc I I . J. ahc _ ^ /rr.\ 



V =z J I sm X =z qt — [cos x^ — cos xj. (52) 



y X 



4 

 The whole volume is V= - Qabc. 



o 



27°. From the equation of the parted hyperboloid are easily and 

 directly obtained 



^QQ'iQ'z ^^ ^'Ji^Q'j'i = SharSc»'V«(T 



= ShxSaVoo' = ShxSa^y = abcShx, 



