32 Proceedings of the lioyal Irish Academy. 



3. Form of the fiuuiion n. 

 The function tt was defined by (9), henee depends on i// - i//'. Now i/- was 

 given by (6) ; to find ,/»' we use (4), thus, 



= S<^a Vpdii, as in (4), 



= - SOfi Vp4>a, identically, 



= - S\W Vp(pa, as in (4) ; (12) 



hence by definition of a conjugate 



^■fi = -ffVp^^. (13) 



Returning to (9) and using the values of ;^/3 and ;//'/3, we have 



«' VpB^ + V Vp^ti = Vnpii. (14) 



which is to be solved for the linear vector function n. Tliere are many ways 

 of solving. We may choose first a method which, while slightly unsym- 

 metrical, brings out the relation of the equation to Joly's invariants, and also 

 has the advantage of compactness. 



Since p may be any vector whatever, write p = fl/3, whence 



e'V,pi5i)fi= V^nefi. (15) 



Multiply the left side by any vector A, and transform thus, 



SWV^ftRfi - SeX^iiOii, as in (4), 

 = .s>/ie/30A, identically, 

 = .S>/3F0/3OA. (16) 



Now, by a well-known relation due to Hamilton, 



VefidX-' md'-'VliX. (17) 



where m is the coefficient of the absolute term in the cubic 



e* - ?n"fl* + m'e - M = 0. (18) 



Therefore (16) becomes 



sxe' f>/3e3 = ms^^W' f/3a 



= »/i.s'd->|3K/3A, as in (4), 



= - 7nSXfie-^<pfi, identically ; (19) 



but, since A was any vector whatever, this is equivalent to 



e' Ff^Oii = - m J73e >f3. (20) 



Tt is worth while to note in passing that (20) is a special case of the identity 



OT^Oii ^-mViiO-'p. (21) 



where nt is the third invariant of 9 and p and /3 are any vectors whatever; 

 this relation may 1>€ proved as in (19), writing p in place of ^/3. 



