ON THE FUNCTION arc sin (x + iy). 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878), 
pp. 171—174.] 
The determination of the function in question, the arc to a given imaginary 
sine, is considered in Cauchy’s Exercises d’Analyse, &c., t. III. (1844), p. 382; but it 
appears, by two hydrodynamical papers by Mr Ferrers and Mr Lamb, Quarterly 
Mathematical Journal, t. xm. (1874), p. 115, and t. xiv. (1875), p. 40, that the question 
is connected with the theory of confocal conics. 
Taking c = J(a' 2 -b 2 ) a positive real quantity which may ultimately be put = 1, 
the question is to find the real quantities £, y, such that 
£ + irj = arc sin - (x + iy), 
c 
or say 
so that 
x + iy — c sin (£ 4- iy), 
x = c sin £ cos iy, iy = c cos £ sin iy. 
It is convenient to remark that if a value of f + iy be % + iy, then the general 
value is 2mTT + f ' 4- iy or (2m + l)7r-(f +iy) ; hence, y may be made positive or 
negative at pleasure ; cos iy is in each case positive, but -r sin iy has the same sign 
as y ; hence cos £ has the same sign as x, but sin £ has the same sign as y or the 
reverse sign, according as y is positive or negative ; for any given values of x and y, 
we obtain, as will appear, determinate positive values of sin 2 £ and cos 2 £ ; and the 
square roots of these must therefore be taken so as to give to sin £, cos £ their 
proper signs respectively.