118 



PROFESSOlt G. H. DAKWIN OX THE INTEGRALS Of THE 



Now we must take for f and g values such as to bring the two definitions into 

 accord. This is the case if 



and / ? </V = 1 + A 

 Hence 



/ x 47rM , 

 (cos) - (1 + 



agreeing with the result on p. 549 of "Harmonics" for the case i 

 type OEU. 



(2) The Scctorial Cosine Harmonic. 

 I define this thus. 



P/ ( M ) = (1 - K 2 sin 2 0)' = f. (K'- + K- cos 2 0)*, 



(7), 

 1, * = 0, 



<, 

 where f =],;/= , . 



s f/> 



= / . K~ K- sir 



By symmetry with the last result 



r , , x 47r/' :i cos /3 cos y ..., :, ,., 4-rrk"' cos fi cos y 

 J 1 1 (cos)= , . 3 / ./-ryV-= . 7 ^ 



J 3sm s /3 3sm 3 p 



In " Harmonics" 1 defined tlie functions thus, 



(9)- 



(i^ 1 (^)) = cos (j) 



If we take f = ( ] +/ ?f, </K' = 



\\ /8/ 



//(cos) = >AI 



. (10). 



J 



I -/3 



definitious agree, and we have 



| ff M (1 + 2)8 + 2/3-) .... (11). 



This agrees with the result on p. 54'J of " Harmonics' with / !,'>= 1, type OOC. 



I define this thus, 



(3) The Sectanal Sine Harmonic. 



, l ( /jl ) = C os - f, K cos 0, "1 



' (<^) = sin rji = >/. i/ I . K' sin (/>, f 



(12), 



where /'= , '/ 



K K\/ 1 



