46 
MR. T. Y. BAKER AND PROF. L. N. G. FILON : LONGITUDINAL 
it follows that 
1 + 4n 0 2 sin 2 a 0 /n 2 2 B.. (36) 
is the required convergency factor. 
If the formulae of § 5 are to get accurately over the failure of convergency, this 
convergency factor should be identical with 
1 + B sin 2 ao/Mj 2 , 
that is, we should have 
B = 4n 0 2 M 1 2 /n 2 2 It,.(37) 
which, when written out, becomes 
B - 4w 0 2 n 2 2 M 1 2 (l—M iYf(n 2 —n 0 ) 2 (1 — 2M,) (n 2 +n 0 — 2n 0 M) (n 2 + n 0 — 2n 2 M 1 ). (38) 
This does not agree with the previously found value for B, being of fractional form 
in M,. It does lead to B becoming infinite of the order when M, tends to infinity, but 
it indicates an infinity of B (and therefore a critical failure of convergency) at three 
other places, namely when M] = \ (n 0 + n 2 )/n 0 , ^ {n 0 + n 2 )/n 2 , at none of which does a 
failure of convergency really occur, as can readily be verified. 
The reason for this is made clearer by geometrical reasoning as follows :— 
Let IP(fig- 3) be a ray which is parallel to the axis in medium 2. To make the 
figure easier and the quantities dealt with positive, the refraction has been taken 
from a denser to a rarer medium, so that n 0 /%> 1. 
Fig. 3. 
In the triangle IqCjPj of fig. 3 we have sin \fs.J sin \[s 0 = I„P,/,^. 
But 
sin G 2 / sin \fs 0 = njn 2 ; hence I (J Pi = n 0 xjn 2 . 
