272 



Mr. W. D. Niven on the Calculation 



quantity K, and therefore 2b, is taken constant, its mean value over the 

 arc as near as it can be guessed at being used. 



Taking this law, the above results, written in the order in which they 

 would be used by a calculator, are as follows : — 



(THTKi^ <•> 



{^H-tHj*- <k> 



'-wfW <* 



X- o f° sec^4 (e) 



The last three integrals Mr. Bashforth has given ample tables for, 

 corresponding to different values of y and between certain ranges of 

 angle. If there is no table for the exact value of y which results from 

 equation (c), a method of interpolation must be employed. The integral 

 P<p is in this case 3 tan + tan 3 0, and its values are tabulated as well as 

 those of its logarithm. 



It will thus be seen that there are six distinct operations of some 

 length, the first being the most serious, because there is some difficulty 

 in getting K right. Supposing, however, that point to be settled — and I 

 shall afterwards offer a few observations which will, I think, make the 

 solution of (a) easier — the quantity y must be found. It will be seen that 

 (b) and (c) are mere stepping-stones to the time- and distance-integrals. 



I shall now enter into an examination of the equations of motion, with 

 the object of proving other formula?, which, when we have once discovered 

 the velocity at the end of the arc, will give the time and distances with 

 two operations. 



Second Method. 



§5. I remark, first, that although Bernoulli's solution succeeds in 

 expressing all the quantities in terms of integrals of <p, yet, owing to the 

 difficulty and complexity of the integrals, it is practically valueless, except 

 in two cases — first when n = l, and next when ?t=3. I doubt, moreover, 

 whether, in the case n^3, it is the best solution when p is not constant, 



