86 
Proceedings of Royal Society of Edinburgh. [sess. 
at least 150° C., and probably meeting the iron line at the neutral 
point 294° C. 
Set. 
t 
(Hot Temp.). 
D 
Observed and 
Corrected. 
D 
Calculated (1). 
D 
Calculated (2). 
A 
37° A C. 
1010 
1006 
A 
41°’6 
1142 
1157 
A 
44°-2 
1238 
1244 
A 
51°*1 
1476 
1476 
B 
* 63° -2 
(1882) 
(1882) 
A 
68°'5 
2066 
2061 
A 
74°-4 
2284 
2259 
2268 
B 
84 o< 0 
2812 
2825 
A 
86° *8 
3009 
2982 
B 
105° ’8 
3948 
3993 
B 
105°‘9 
4029 
3999 
A 
107°*4 
4041 
4075 
A 
108°*7 
4155 
4140 
B 
113°*2 
4324 
4362 
C 
ni4°-o 
(4400) 
(4400) 
C 
127°-5 
5018 
5030 
C 
138°-4 
5485 
5499 
c 
151°'0 
6025 
6008 
Note on the Relation between the Mutual Distances of 
Five Points in Space. By Thomas Muir, LL.D. 
(Read March 4, 1889.) 
Lagrange, in his paper “ Solutions analytiques de quelques prob- 
leraes sur les pyramides triangulaires,” Nouv. Mem. de V Acad. 
Roy. . . . (de Berlin), Ann. 1773, pp. 149-176, gives unintention- 
ally the following expression of the relationship between the mutual 
distances of five points in space, viz. — 
4A 2 /= a(a +/- <jf + a (a +/- g'f + a" (a" +/- <j'f 
+ 2/8(a' +/ - g')(a" +/- g") + 2/5 \cc +f-g)(a" +f-g") 
+ 2/5"(a +/- g)(a' +/- g ') , 
where, if the points be called 1, 2, 3, 4, 5, the letters 
f 12 13, 14, 
represent the squares 23, 24, 
of the distances 1 34 , 
c a 
c a 
a' 1 
i- 
/ J 
15 
25 
35 
45, 
and A, a, a, a", ft, ft, /3", are certain functions of the said distances. 
