210 Of the SOLIDS 



Therefore (i — JL^ • (f +/)V = * _ ^ . _Z_, or 



(2 # — j) (y+/) 2 = 2# a — j/ 2 . 



As this equation is homogeneous, if we make <- =:*/, or 

 y — u x, both a? and j/ may be exterminated. For we have by 

 fubftituting u x for y, (2 x — u x) (x 2 -\-u z x*) a = 2 ** — u 2 w\ 

 or (2 x 2 — u x*) (1 -f- «*) 2 = 2 # a — a 2 **, and dividing by x z , 



(2 — »)»(i-f-# 2 ) 2 s? 2 — u * \ whence fquaring both fides, 

 (4 — 4« + « a ) (i+» a ) = 4 — 4"* + « 4 - 



From this, by multiplying and reducing, we get 4 a 1 — - o«. 



— — 4 ? or u 1 — ^ u =. — 1; and // z= 9 — ^ ' .. 



4 ». 



2. The two values of « in this formula create an ambiguity 

 which cannot be removed without fome farther inveftigation. 

 If A be the attraction of the cylinder, then A = 2* (x -\-y — 



v / (I .' i _j_y'*) ) into which expreflion, if we introduce u, and exter- 

 minate both x andj/, by help of the equations # xy x — m 3 3 and 



v A t i+« — v/i + «* 

 — w, we get A= 2.^3 m — : 



2 



Notwithstanding the radical fign in this formula, there 

 is but one value of A, correfponding to each value of u, as the 



pofitive root of s/i — u x is not applicable to the phyiical pro- 

 blem. 



