the Five Aberrations of von Seidel. 679 
instrument are included in the form of the characteristic 
function V, or in the form of this other connected function, 
T=te+f»y + n»-ZV~my-nV— V, . . (3) 
which may be considered as depending on only three inde- 
pendent variables besides the colour ; namely, on the incli- 
nations of the final and initial portions of a luminous path to 
each other and to the axis of the instrument. Algebraically, 
T is in general a function of the colour and of the three 
quantities, l 2 + m 2 , lV + mm\ l /2 + m /2 ; and it may usually 
(though not in every case) be developed according to 
ascending powers, positive and integer, of these three latter 
quantities,- which in most applications are small, of the order 
of the squares of the inclinations. We may therefore in most 
cases confine ourselves to an approximate expression of the 
form 
T = T (o ) + T (2) + T (4) 5 ( 4) 
in which T (0) is independent of the inclinations ; T (2) is small 
of the second order, if those inclinations be small, and is of 
the form 
T( 2 > = P(Z 2 + 77* 2 ) +P 1 (ZZ / + mm') + P'(Z' 2 + m^) ; . (5) 
and T (4) is small of the fourth order, and is of the form 
T« = Q(l 2 + m 2 ) 2 + Qi (I 2 + m 2 ) (IV + mm') 
+ Q'Q 2 + m 2 ) [V 2 + m' 2 ) + QnQl' + mm') 2 
+ Q 1 '(ll'+mm f )(V 2 + m' 2 ) + Q"(r 2 + m' 2 ) 2 ; . (6) 
the nine coefficients, P P x P' Q Q ± Q' Q u Q x ' Q", being either 
constant, or at least only functions of the colour. The 
optical properties of the instrument, to a great degree of 
approximation, depend usually on these nine coefficients and 
on their chromatic variations, because the function T may in 
most cases be very approximately expressed by them, and 
because the fundamental equations (1), (2) may rigorously 
be thus transformed ; 
__l_ _^T m dT m ^ 
n ell ' n ~ dm ' ! 
,_^,__^T m' ,_ dT | " {i) 
X n' Z ~ dl" V n' Z ~~ dm!' ) 
The first three coefficients, P P x P', which enter by (5) 
into the expression of the term T (2) , are those on which the 
focal lengths, the magnifying powers, and the chromatic 
aberrations depend : the spherical aberrations, whether for 
2 Z2 
