8 Proceedings of the Pvoi/al Irish Academy. 



tlie integral considered by Hamilton is 



Q= pi^(^, d^), (2) 



in Tvidch. the variable quaternion q changes by a determinate mode 

 of passage from one fixed limit g'o to tbe otber q-^ . He supposes tbe 

 mode of passage to undergo a slight rariation while the limits remain 

 fixed, and he denotes the corresponding variation in the integral by 



SQ = Sf' F{q,diq) = [' hF{q,dif). (3) 



l^OW 



^F{q,d.q) = S,Fiq,dq) + F{q,8clq) (4) 



in -which S^ is a symbol of partial differentiation and relates to 

 q alone and not to S^ . Similarly 



dF{q, Sq) = d,F{q, Sq) + F{q, dSq) , (5) 



and because the differentials dq and Sq are independent 



8dq = dSq. (6) 



Therefore, subtracting (5) from (4) we find 



8F{q, dq) - dF{q, 8q) = 8,F{q, dq) - d,F{q, 8q) , (7) 



and when we integxate this between the fixed limits we obtain 

 Hamilton's result 



8Q = \''{8^F{q,dq)-d,F{q,Sq)}, (8) 



because 8q vanishes at the limits. 



Art. 2. — Hamilton contents himself with observing that the 

 elements of the integral (8) do not generally vanish, and therefore 

 the value of the integral (2) depends in general on the mode of 

 passage. We shall suppose that it is possible to pass by continuous 

 variation of the mode of passage fi'om one given mode to another 

 without the introduction of infinite terms. In this case we shall 



