ME. R. A. SAMPSON: A NEW TREATMENT OF OPTICAL ABERRATIONS. 1.51 
then, provided we do not change the order in which the sclieines ])resent tliemselves, 
the composition may ])e effected in such groups as may be convenient, and may be 
performed either from left to riglit or from right to left.* 
The product of the determinants of the component sclieme — ••• 
gives tlie value of tlie determinant GL —HK of the compounded scheme. 
The analytical scheine corresponding to any instrument of which the cardinal 
points are known may be written down at siglit, and conversely, by the relations 
O F H N * O' h' n' F' 
Fig. 1. 
OF = ///.-, HF = nik, NF = ilk ; O'F = -gjk, H'F' = -ijk, N'F' = -nlk, 
where 
n = yl-hk .(4) 
and H, H', N, N', F, F' denote, as usual, the unit points, nodal points, and principal 
foci respectively. 
If it is desired to work geometrically, ^we may set the original and emergent axes 
across one another in a figure at any angle, H and H' being superposed (or else 
p\ 
N, N'), and finding a point 0 with co-ordinates HF, HT'; then any straight line 
fiPP' through 0 determines points PP', which are conjugate foci. 
The following method of compounding any two given systems may also be 
mentioned :— 
* A general discussion of the linear system by the author ■will be found in ‘ Proe. London Math. Soc.,’ 
vol. 29, p. 33. 
