C. S. Hastings—Oolor Correction of Double Objectives. 171 
hence the total quantity of light falling upon @ would be equal 
to the sum of three integrals, the first taken from the extreme 
red up to n_,, the second from n_, up to n,,, and the third 
from this last limit up to the extreme violet; that is, if g be 
this quantity, 
fi, dn f ee ae a ae 
dat erregen ees 1,0n+ af 1.3 pie" 
q ny (Pee Vk / ie (P)—P,)? 
To find the value of mp which will render the last expression a 
maximum, it is necessary to find the first derivative of the 
function with respect to my and set it equal to zero. Remember- 
ing that need, we see that P,, n_, and n,, are the only quan- 
0 
tities dependent upon 7, hence we obtain by differentiation 
under the signs of integration, 
2 sane 2 sf 1,0 : dl Gr le tk oa 
iN psepettriepadt 2 {P.—P.Y dng (P.—Pp_,)? 
1,0” d oe 
n : bees 
Inga in 208 , (Po=P.) day (P—Pay)® 
But from the definition of the quantities n_, and n,4, we have 
(Po —Pn_.)? =(Po— Pn)? =2?. 
Hence the value of a reduces to minus the product of 2a? 
Ne 
into the sum of two definite integrals, and the problem becomes 
simply the determination of the value of , which will cause 
this sum to vanish, or, since must have opposite 
dP, 1 
dn, (Po—P,)? 
signs in the two integrals because ioe that value which will 
0 
make the two integrals equal. If both z, and P,, were sym- 
metrical with respect to some one value of n, that value would 
evidently answer the condition; otherwise it is necessary to 
know the function ¢,. 
It would be easy to find a transcendental function which 
would express empirically the photometric observations of 
Fraunhofer, or the more recent and elaborate ones of Vierordt, 
but a simple consideration of the physical limitations of the 
problem will enable us to dispense with this operation. 
Hitherto we have made no restrictions as to the value of a, 
Let us now suppose that a is so small that it may be regarded 
as the image of a star, then a is small and, since ae =0, P, may 
