82 Mr. J. F. W. Herschel on the action of 
of their respective curves. The numbers in the 5th column 
exhibit the excesses of the terms of the arithmetical progres- 
sion in the 4th (whose common difference is 1*59, the mean 
of all the differences in the third column) above the observed 
values of a b, and are so small as fully to authorize the con- 
clusion, that these values, and of course those of the para- 
meter b, increase in arithmetical progression with the order 
of the rings ; or in other words, that the number of periods 
performed in a given space (=1) by a luminous molecule 
going to form any point M in the projection of any ring, is 
proportional to the rectangle of the distances P M, P' M of 
that point from the two poles. 
Now, if we extend our views to crystals in which the dis- 
tance between the axes is considerable, we may reasonably 
expect that the usual transition which takes place in analytical 
formula? from the arc to its sine, when we pass from a plane 
to a spherical surface, will hold good. If this be the case, we 
shall have at once, and in all cases 
4 ( 9 , 9 ') = sin 9 . sin 9 ' 
and the nature of the isochromatic curve for the n th com- 
plete period will be expressed by the equation 
sin 9 . sin 0'= j~ t ' cos <P = nh . cos <p ( e ) 
putting h for ^ • If the plate be cut at right angles to the 
optic axis 
cos 6 + cos S' 
COS <p — 
T 2 . cos a 
and consequently 
sin 9 . sin 8' = (cos 9 + cos 9') ; (f) 
To put this to the trial, I took a plate of mica, whose thick- 
ness measured 0-023078 inch, and having adjusted it accu- 
rately on a divided apparatus, placed it in an azimuth 45 0 , 
