14 REPORTS ON THE STATE OF SCIENCE.—1918. 
For greater values of A PR, may come by either of two paths. It 
is easy to take from a diagram the alternatives as follows :-— 
° ° ° ° ° ° fo} 
AS. : = 24 25 2 27 28 29 30 
Are (A—®) . 6-0 4:5 40 36 3:3 30 2°7 
Gong are. . 18:0 205 22-0 23-4 24-7 26:0 27°3 
ne 8. S. s. cA 3. ret, 
4(S+P)—PR, . 107 113 118 122 127 133 138 
fe} (eo) °o ° Oo fe} Cc 
Are (A+). Spel) 8-0 8:9 9-7 10-5 11:3 120 
Long are. -, 180 17:0 17-1 17-3 175 17-7 180 
8. s. s. s. 8. s. 8. 
3(S+P)—PR, , 107 104 104 105 106 107 150 
Here it certainly looks as though the observed PR, starts with the 
longer arc A+¢ instead of the short arc A—d¢ as we have assumed 
for PR;; for only by this supposition can we obtain approximate con- 
stancy for the quantity }(S+P)—PR,. There is nothing unreasonable 
in this difference between the two cases: for PR;, if we are right in 
identifying it with Y, is usually read for 8; we read the first big move- 
ment, and, as already remarked, the PR; by A+¢ follows that by 
A—g¢. Hence the former is read. But in looking for PR, we should 
naturally take a movement which is not too near P. The PR, which 
starts with A—@ runs up closer and closer to P as ¢ increases. 
We have, however, still to explain why (S+P)/2—PR, should be 
about 105s. instead of about 80s., as at the head of the * observed ’ column 
in Table IIT. On the present hypothesis the tables are wrong to this 
extent. When A 28°, for instance, if we take the last column of Table 
VI. the error of P is —12s. and that of S should be nearly double, say —20s., 
which gives for (S+P)/2 an error of —16s.: and PR, isin error by 2 x 14 
=28s.; making altogether 44s., which reduces the 106s. of the tables 
to 62s., as compared with 75s. observed. These correctionsare apparently 
too large, and it may be readily admitted that the last column of Table 
V1. probably goes too far.” If we use the first column we get, say, —8s. 
tor (S+P)/2, and —22s. for PR,, which is just the quantity required. ~ 
Somewhat similar considerations apply to PR,. The whole are is 
now 
and becomes a minimum when 
d¢=bddA, 
which leads to 
(5?—1) sin? d=tan? C. 
so that @=1°-7, A=8°-75: minimum A=42°. 
Generally, for PRn, the corresponding equation is 
{(2n+1)? —1} sin’d=tan? C, 
or (2n+1)¢=C approximately ; so that when n=5, d=0°-77, A=8° 58; 
minimum are being 93°-6, close tothe value obtained by starting along EC. 
