1893.] On Operators in Physical Mathematics. 127 



-'-r-l). (87) 



Integrating, we obtain 



-r-i). (88) 



This is quite correct when r = 0, when we obtain the ordinary 

 formula 1 0" + ..... Another form of (88) is 



. (89) 



Now when r = J, the square of the sine equals unity throughout, 

 giving 



'ar*+ (jf )'arl + (|f )'ari + ..... . (90) 



Since we also have 



aji ajt a i 



the product of (90) and (91) should be unity. That is, 



bi)' (i-*)' (bay d-H)' 

 : ~ 



+ I4+ 14 + 14 + ----} 



, , 



f+ 



+ ..... (92) 



Going by the ordinary principles of the algebra of convergent 

 series, we should conclude that the coefficient of # was 7r 3 , and that 

 the coefficients of the other powers of x were zero. But this rule is 

 not generally true in series of the present kind, as we have already 

 exemplified. Therefore, to see how it goes in the immediate case, I 

 have calculated the value of the coefficient of x. By (84) we have 



