WITH AN APPENDIX BY PROFESSOR A. LODGE. 107 



In this respect LAPLACE'S approximate formula for high values of , viz. : 



P,, (ff) = 



- 

 y (mr sm 6) 



cos nd 



4 



shows a wonderful resemblance to the actiial functions even for quite low values of n. 

 The numerical values of this function are, indeed, not very near the true values 

 even when n = 20, as will be seen by the following short table : 



But, though its numerical values are not very close, the positions of most of its 

 zeroes are remarkably near the correct places. It can, of course, only be considered 

 between and 180, since sin 6 becomes negative beyond these limits. But 

 between these limits it has n zeroes, with n 1 equal intervals between them, the 

 first zero being at 9 = 270 -=- (2n + 1), and the interval between successive 

 zeroes being 360 -f- (2n + 1), the formula for the required values of 9 being 

 9 (4r + 3) 90 -f- (2n + 1), for integer values of r from to n -- 1 , 



Taking n = 20, this would make the first zero approximately at G 35', and the 

 constant interval 8 47', very nearly; the roots given by the formula being, roughly, 



6 35', 15 22', 24 9', 32 55^', 41 42-', 50 29', 59 16', . . . 



The actual value of the first root of P.-, (#) = is slightly over f> 43', and intervals 

 between successive roots are very nearly equal, varying between 8 43' and 8 47'. 



The first ten roots are, to something like the nearest minute, 6 43', 15 20', 

 24 11', 32 57', 41 44', 50 30', 59 10', 08 3', 70 50', and 85 37'. 



Professor PERRY has brought out a table of Zonal Harmonics to 4 decimals, for 

 every degree, as far as P 7 (' Phil. Mag.,' December, 1891), and by help of this table 

 I have calculated the first root, and the intervals between successive roots, for P 3 to 

 P 7 , to something like 1 minute accuracy. Their values, and the corresponding 

 approximations obtained from LAPLACE'S formula above, are given in the following 

 table, showing how far they differ for these low values of n : 



P 2 



