7,0 PROCEEDINGS OF SECTIOK A. 



5. If we now substitute the vector expressions from sections 

 2, 3, 4, in equations I., and equate separately to zero each set of 

 vectors terms of the same order, we obtain the two series of vector 

 equations — 



ra, + 20) .^<j /a^ + -^ („^^ _ -, + a, + i) j- =. 0, (IV.) 



where q is any odd number — 



pap + ;?w t - .j Xa^ + -^ (a^ _ i + a^ + i) 1=0, (V.) 



where p is any even number except zero, together with ^ = 2?;/^. 

 Prom lY. we get a series of equations of the type— 



* ae,-\ + 2 i a,; + a,, + i = 0, 



qixyin 



or — 



«(/-! + t,,-An + aq + I =0, 



where tq is the operator T>,,l '' , iu which — 



J), cos fq = 2— , D,, sin/^ = — ^, 

 in quim 



that is — 



'm-\ q-(i)- / qwl 



Similarly from V. we get the series — 



a^;-! + T^,aj, + {\p + l = 0, 



where Tp is the operator A^,*. ^p , in which — 



Aj3 cos ({)p = 2 - , Ap sin <fip = ^^ 

 VI pwm 



^l = %(x^ + -^ Y tan c),p= -P-. 

 oil' \ pw / p(x}K 



Note that the vector equations in this paragraph are equations 

 connecting the different vectors, considered purely as vectors, with- 

 out any reference whatever to the order of harmonic they originally 

 represented. 



6. We have thus obtained the following infinite series of equations 

 connecting the vectors used to represent x and f : — 



^.^i + «. = — a^ 



a, +■ T,a, + .Mj =: 



a, + /,a, + a^ = (VI.) 



a, + T^a^ + a^ =0 



tt^ + ^^a, + a = 

 &C., &C. ; 



