MATHEMATICAL SECTION. 157 
This value of ¢, makes the co-efficient of y in the first normal 
equation and the co-efficient of z, in the second normal equation, 
zero, and hence gives directly 
[pn] 
yornv a 
_ [pt —t,)n] 
~ [pé—47] 
The weight of this value of x, is a maximum; 7. e., the value of 
pt] 
x, corresponding to ¢, = 7 P| has a greater weight than the value of 
x, corresponding to any other value of t,. 
The probable error of the function x, + yy is given by the simple 
formula. 
V ot wre, 
in which ¢,, and «, are the probable errors of x, and y, respectively. 
The investigation shows that, when several standards of length 
are to be intercompared two and two, in order to obtain the length 
of some one of them, it will be conducive to accuracy to have the 
mean temperatures of the several sets of comparisons equal. 
Remarks were made upon this communication by Mr. KuMMELL. 
Mr. Avex. S. CHRISTIE made a communication on 
CONTACT OF PLANE CURVES.* 
[ Abstract. ] 
Let 0 = fiz, y), 1), 0 = ¢(@, y), (2), and y = ¢(2), (8) be the 
equations of plane curves. Transferring the origin to (¢, 7), where 
= $(&), writing f, ¢ for f(§, 7), ¢(§, 7), respectively, and u, for 
1 of 1 og 
dee? Yn for — I =e , we have 
y" o2 *o 
from (1), ae (ES iy 73 (oru,)), (1’), from (2), 
2 3 30 a dy 
leap} oak 2(a'v,), (2'), and from (3), y= # z pty ge 
S| 
Br age + &e. ve 
* Throughout this paper, d, for lack of sorts, is put for round d, and denotes 
partial differentiation. 
