462 



" Referring to the memoir,* we find, adopting the nota- 

 tion so often described, 



P + p = \/{ (« + ''«! + ^«-^ + '^"3)2 + {b + G/3i + b^.2 + c(5,y 1 

 ^ I + {c + ayi + by.2 + Cyzf. ] 



Assuming now 



X = ai^ + /3i' + yx\ ^ = a^^ + (3.,^ + yi, v = a,^ + /Ba^ + 73^, 

 ^ = 0203+^32/33 + 7273? X = "3«l+/33^1 + 737l5 l/' = aia2 + /3l)32+7172» 



and developing the square root, we obtain : 

 P + p = 



1 + (oi cos2cr + (8.. COS2/3 + 73 cos^y + Mcos/3 COS7 + ucosy cosa + wcosa cos/3) 



{■ 



I (\cos2a + iicos-fi + I'cos^y + •20cos^cos-y + 2xcosycosa + 2)pcosacos/3) \ (2) 

 (aicos^a + /3ocos2/3 + yacos^y 4- «cos/3cosy + Dcosycosa + u;cosacos/3)2 J 



neglecting terms of a higher order than the second. 



" The function arising from this expansion will be (vid. 

 Transactions of the Academy, vol. xxi. p. 153) : 



2 F= 2 ( Gai + H(5., + /7;j) + Z)m + Ev + Ftv ^ 



+ {GX + Hfi + Iv +D(p + Ex + F\p) [. (3) 



+ 2^ J 



where <P denotes function (1), • 



2 G = \\lF,p COS^af/w 2 ia^ = jT|T,p cos2|3f/a; 

 21= Jjl i'op cos-7(Zw 

 2> = l\lFop cosj3 C0S7JW -E" = .[[f-fo/o COS7 cosaf/w 

 2^= JjJ-Fojo cosa cosj3f/w, 



rfw being the element of the volume. 



" This function (3) contains twenty-one coefficients, and is 

 quite distinct from the function which may be derived from <1>, 

 by introducing arbitrary coefficients. If the terms G, H, I, D, 

 E, F, be retained, the natural state of the body will not be one 

 of free equilibrium, and the equations of a homogeneous body 



• Transactions of the Royal Irish Acadpray, vol. xxi. p. 153. 



