176 Mr. A. W. Porter on the Diffraction Images 
Gi.) When sin v=0 the differential coefficient does not 
equal infinity but zero. The approximate value when v 
approaches an integral multiple of 7 is 
dk, _gv(l—n'ja 
dv Bat 
where a=v—pm7 (p=integer); and this obviously vanishes 
when a is zero. Since n?>1 itis positive when a is negative, 
and negative when a is positive ; the vanishing of a therefore 
corresponds to a maximum of F°?. 
ProBLEM B.— The relative intensity of the principal and 
secondary spectra of a diffraction-grating. 
Though this question is discussed with sufficient approxima- 
tion in Schwerd, Beugungserscheinungen, some misconception 
seems to exist in respect to the degree in which an increase in 
the number of openings in a erating diminishes the intensity 
of the secondary maxima compared with the principal maxima. 
For example, Mascart in his Trazté (t. 1. p. 355) makes the 
following statement :—“ Tant que sin’v différe notablement 
de zéro, les maxima secondaires sont tres faibles par rapport 
aux Maxima principaux ; ces maxima ont leurs moindres 
valeurs au milieu de Pintervalle des maxima principaux et 
augmentent & mesure qu’ils s’en rapprochent. Dans tous les 
cas, le rapport...des maxima secondaires aux maxima 
principaux est d’ autant plus petit que le nombre N des ouver- 
tures est plus grand, et ces maxima finissent alors par devenir 
insensibles.” 
Again, Drude in his ‘ Theory of Optics’ (p. 223) states: — 
“Between the fringes [7. e. the secondary minima] the in- 
tensity J reaches maxima which, however, are at most equal 
to the intensities produced at the same points by a single slit.” 
How far these statements are true may be ascertained as 
follows :— | 
The secondary maxima are given by those values of v for 
which 
cot De 
ol) = 
Tr 
o % ie 
and .’. sin? nv= : 
n? + cot? 2 
Their intensity is therefore 
. 2 2 
r= sin nv <2. led ll Aca 
7eReces ee) Nn Coe ee nS 
sin’ v nm sin* v+cos*2 
n? 
~ 14 1) sin? v* 
