278 



Mr. W. D. Niven on the Calculation 



It thus appears that when n=3, 



2 (a-© 2 

 X l + cos(a+/3) 3 ' 



and 



aY___ 2 tana (a-/3) 2 

 Y sin(a+/3) 3 ' 



Now suppose we were integrating oyer an arc of 5°, then ( a ~^) 



o 



might, approximately, be put equal to 3X |g 81 = j^' O ur results would 

 therefore be less than -^wu m error. Moreover the error in T, which 

 is really the more important of the two, is less than this, as I have 

 pointed out at the beginning of this article. If, therefore, the formulas 

 are otherwise serviceable, their inherent errors do not seem to be a great 

 objection to their use. 



§8. The formulae for X and T already found apply only to the 

 ascending branch of the trajectory. A little consideration enables us to 

 see that the same formulae apply to the descending branch, provided the 

 mean angle (j> is 



a + /3 _ p — q ft— a 



2 0~2)p-0-3)g~6~' 



ft being now greater than a. 



§ 9. In the preceding articles we have neglected all consideration of 

 the time-integral. It may, of course, be treated in the same way as the 

 distance -integrals. I shall not, however, go into the general case, but 

 shall merely state the result in the important case when w=3, 



rp _ cos" 0' 4 ^ d u 



/ r p ch 

 1 



where <j> is equal to 



q+/3 , p-q (q-/3) 

 2 ^9p~3q 2 



in the ascending branch, and 



« + p — q (ft — a) 

 2 9p-3q 2 



in the descending. 



If the arc integrated over is small, 0' is not very different from 0. 



§ 10. I propose now to collect the results I have proved, stating them 

 in the order and form in which they would be used in conjunction with 

 Mr. Bashforth's tables for K. 



