[34 PEOFESSOR G. H. DARWIN ON THE INTEGRALS OF THE 



It may be convenient, as furnishing a kind of index to the foregoing investigation, 

 to state that the 1 + 3 + 5 + 7 integrals for the harmonics of orders 0, 1, 2, 3 are 

 given in equations 3, 5, 9, 13, 16, 24, 28, 32, 38, 47, 56, 64, corresponding to the 

 definitions contained in 4, 8, 12, 15, 23, 27, 31, 36, 45, 54, 63. 



The definitions of the harmonic functions as given in my paper on Harmonic 

 Analysis are repeated in 6, 10, 19, 21, 25, 29, 34, 39, 40, 42, 43, 48, 49, 51, 52, 57, 

 58, 60, 61, 65, 66. Corresponding to these latter definitions the approximate 

 integrals are given in 7, 11, 14, 20, 22, 26, 30, 35, 41, 44, 50, 53, 59, 62, 67 ; and the 

 results confirm the correctness of the general approximate formulae for the integrals 

 given in 22 of the paper on Plarmonic Analysis. 



A mistake in that paper was detected in the value of the cosine-function for the 

 third zonal harmonic, and the corrected value is given in (40). 



It must be obvious that the method exhibited here may be applied to higher 

 harmonics with whatever degree of accuracy is desired ; but it is also clear that the 

 labour of evaluating the integrals increases very much as they rise in order. It is 

 probable that the approximate results of the previous paper will suffice for most 

 practical applications. 



APPENDIX. 



On the Symmetry of the Cosine and Sine-functions ivith the P-functions. 



In my previous papers I fiiiled to notice that the symmetry between the P-functions and the cosine and 

 sine-functions is not destroyed, but is only masked, in the approximate expressions for the harmonic 



functions. 



For example, (39) shows us that 



therefore, in consequence of the symmetry which subsists, we ought to find 



G,W=P. [( 'f 2 *)' 



Whence 



Cs (</>) = -7?-4dP^ (1 - ft cos 2<f>) (1 - /? cos 



This only differs by a constant factor from the expression (40). 



It would be possible then to have only one type of function, viz. $ or p, and to express all the cosine 

 and sine-functions by means of the appropriate one of them. This would be found to be equivalent to 

 expressing the latter functions in terms of powers of sin <. For the purposes of practical application I 

 do not think this would be so convenient as the use of cosines and sines of multiples of <f>, and the 



