150 MR. R. A. SAMPSON: A NEW TREATMENT OF OPTICAL ABERRATIONS. 
"ivhere </, /?, I are constants involving the curvatures of the refracting surfaces, the 
distances l^etween them and the refractive indices; also 
gl — hh = fij/ j! . 
Following Seidi:l, we shall call such systems normal systems. 
In particular, for a single refracting surface, 
2a' = B (?/ + 2:^)+ ..., 
without change of origin, the scheme 
as I shall call it, becomes 
where * is put in place of zero. Or again, a simple shift of origin by a distance d 
may be represented by the scheme 
1, d 
* 
If two instruments be represented by the schemes 
light passing through (l) first and then through (2), and the emergent origin for the 
first being made the same as the original origin of the second, their combined effect is 
given by the scheme 
(3) 
which may be written down by multiplying the rows of the later scheme into the 
columns of the former, as if they were determinants. It will be shown hereafter that 
this nde is remarkably well adapted for numerical calculation—a fact that does not 
seem to have ])een remarked before. The scheme corresponding to any system, as, for 
example, any thick lenses, arranged at intervals along an axis, may be built up from 
its elements by this rule, by writing down the schemes in order belonging to the 
successive refracting surfaces and shifts of origin, and compounding these ; if we have 
to compound in this manner a secpience of schemes 
• • • 5 1 0ni • • • IJ 
