488 Mr. A. M c Aulay on Quaternions as a 



or 



0A=I{(D f P + D,N + D tf M)-(P-Q)^-(P-B)^, 



-N(2,^ + , Wf )-M(2^ + ^ f )}+J{} + K{} . (28) 



Substituting this value of </>A and the values of F and e 

 given by equations (22) and (23) in equation (17) we get 

 equation (19). 



As a second example in curvilinears let us find the strain 

 in this notation. Let ty be the pure strain due to e, i. e. let 



2^©=-Sfi>v- e -ViSfi>ei (29) 



With the usual notation (<?, /, g, a, b 1 c) for the coordinates of 

 pure strain, we have 



2^I = 2*I + cJ+&K, &c, &c. . . . (30) 

 Substituting in equation (29) for y from equation (24), we get 



2^I = D l e-ISI%-JSID 7 ,e-KSID< r e. 

 But [equation (23)] 



D|e = ID f t* + JD f i> + KD f to + wD f I + «D f J + wB^K 

 = I(DfM— r ^0— ^^)+J(D^ + „ot^)+K(D|10 + ^^) [eq. (26)] 



Similarly, 



— SID >) e= D^m + £^0, — SID^e = J)(u + |^i0. 

 Hence 



whence from equation (30) 



Compare this with § 234 of Ibbetson's ' Elasticity.' 



2/=..., 2g=... \ 

 )s+e*t)v, 26=..., 2c = ...j * 



III. Jacobi's Theorem. 



In Todhunter's ' History of the Calculus of Variations ' 

 (§ 323), this theorem is thus enunciated. 



Let v be any function of a, ?/, z, and G any function of 

 #, y, z, 0, ~dvfda;, 'dv/'dy, "dvfyz. Let x, y, 2 be three functions 

 of any three other variables X, /n, v. When expressed in 

 terms of X, /x, v, let v be denoted by <p ; and when expressed 

 in terms of X, fi, v, $, "d<j>fd\ } "dcpfd/u,, "d4>/"dv, let G be de- 

 noted by I\ Lastly, let 



B(x,/A ; v)' 



