formed by a Plane Diffraction- Grating. 175 
coefficient is 
dk? sin nv & COs nV SIN V—siN nv COS ay 
we. SIN YU sin? v bs 
The principal maxima are given by sin v=0, which is said to 
make the coefficient discontinuous (2.9. Jamin et Bouty) ; 
the secondary maxima are said to be given by the equation 
tan nv=ntanv; and the secondary minima by sinnv=0 
(unless sin v=0). The object of this note is to point out that 
sin v=0 makes the coefficient zerv and not discontinuous, and 
that the condition given universally for the secondary maxima 
is incomplete. 
(i.) Take the condition for the secondary maxima first. In 
reality, when x is odd the condition given is incomplete, for 
it does not include the maximum midway between two 
principal maxima. For example, when n=3, although ther 
is a secondary maximum whenever v=odd multiple of = there 
is no value of v satisfying 
3 tan v=tan 3v, 
except even multiples of 7/2, which give the principal 
maxima. 
[In the neighbourhood of v= sit should be noticed that 
3 tan v=9 tan 3v.] 
The fact is the differential coefficient also vanishes when 
cos nv and cos v are simultaneously zero, and this condition 
corresponds to the neglected maxima. 
The conditions can, however, be more neatly stated if the 
differential coefficient ie sentenen in one of the two following 
forms: 
dF _ sin ney 
Jaane (n—1) sin(n+1)v—(n+1) sin(n—1)v] (2) 
a cmconmm— Coney. g4 i. « = os 6). (8) 
The coefficient can be proved to vanish whenever 
(n—1) sin (n+ 1)vo—(n+1) sin (n—1)v=0, 
and this gives all the maxima (principal and secondary). It can 
also be proved to vanish whenever 
n cot nv—cotv=(, 
and this again gives all the maxima. 
The former is more convenient for graphical representa- 
tions, because no infinite values occur. The latter condition 
is perhaps the more convenient generally. 
