

WITH AN APPENDIX BY PROFESSOR A. LODGE. 103 



In these ways all the separate terms were ensured to be free from errors due to 

 carelessness in taking proportional parts, or any other incidental errors. 



Then the terms were added together for each value of in P 2/1 (6} for a given value 

 of n, so obtaining the values required for the actual table. By adding I mean to 

 include also subtracting, the artifice of putting positive terms in black and negative 

 terms in red being a great help in this part of the work. Errors in this work were 

 corrected by adding all the values of P 2l , (6} from 6 = 5 to 6 = 90 for a given value 

 of , and comparing the result with the sum obtained in a different way (see note at 

 the end of the second auxiliary table appended). Up to P 1;J the additions and 

 subtractions and checkings were all done without mechanical aid, but for the later 

 values of i), from P u to P 2u , I made use of an EDMONDSON'S calculating machine which 

 was very kindly lent to me by Professor McLEOD. 



In this way all the even harmonics were calculated and were ensured to be free 

 from errors, except those incidental to the last figure, which is, of. course, only 

 approximate, as the terms used in the calculation were evaluated to 7 decimals only. 

 I am confident, however, that the last figure is never far from the real value, and 

 that it would be more accurate in every case to retain it in numerical work with the 

 tables than to omit it. The error is not usually more than i '2 in the 7th place, 

 and I am confident that it never exceeds ^b 3, whereas omitting it would lead to a 

 possibility of ^ 5 m Addition to its actual error, i.e., to a maximum error of i 8. 

 I have assumed that there are very few numerical calculations requiring an accuracy 

 greater than an approximate 7th decimal place, and that, therefore, the vastly increased 

 difficulty which would have been caused by working throughout witli 8 decimals 

 would have been wasted labour. 



Calculation of the Odd Orders, and Final Checking. 



When the even orders were calculated, the question arose as to the be.st way of 

 calculating the odd orders. P,, of course, gave no difficulty, being merely cos 6. P 3 , 

 also, was quite easy to calculate directly from its value -g- (3 cos 6 -f- 5 cos 3$), 

 EDMONDSON'S machine being used for the purpose. The remaining odd functions 

 were calculated from the even ones by means of the identity 



(In - 1) ?,?_, = nP a + (n - 1) P.... 



The accuracy of the results was checked by recalculating the even P's from the odd 

 ones by means of the same formula. This clinched everything. 



The mode of using this formula which I adopted, between 6 = 5 and 6 = 60 

 inclusive, was different from that adopted between 6 = 65 and 6 = 85 inclusive, so 

 as to minimize the effect of 7th-figure inaccuracies as much as possible. 



