ON THE THEORY NUMBRES. 508 
Meteorological Observations made at different Stations in connexion with 
the Balloon Ascent on 
September 8 (continued). 
Rowat Opservatory, GREENWICH. 
Reading of Temp.| Ten- | Degree Sly 
5 : | Direce- | 27 | 2 
Sala Barom.| Thermom. ee ae Humi- tion of | BE | 8 ¢ oo 
reduced point.| pour.| dity, | Wind. Bale 8 
to 32°F.) Dry, | Wet. <250|<0 
h m in. 3 * O in. 
5 Op.m.) 29°92 | 66:4 | 63°2 | 60°6 |*529 | 83 | sw. | ro | ... Overcast. Balloon disappeared 
behind clouds at 45 55™. 
Saw the balloon due S., mo- 
ving towards Eltham. 
5 10 5, |29°92 | 66°7 | 63:0 | 6o'r |*520 | 80 9 Balloon seen for the last time. 
Overcast; cirrostratus. 
5 20 5, | 29°92 | 65°7 | 63°0 | 60°9 |°631 | 84 9 Clouds broken in S. and W.; 
cirrocumulus. 
5 3° », | 29°92 | 65°6 | 63°0 | 610 | "537 | 85 | s.w. 3 
5 4° » |29°92 | 65:2 | 62°8 | G0°7 | °531 | 85 8 | 
5 50 » |29°92 | 64°83 | 6274 | 60°4 | 526 | 86 oa 6 Cirrocumulus. 
6 © 5, |29°92 | 64°7 | 62°2 | 60'2 |*522'| 86 | sw. | 6 
6 10 5, | 29°92 | 64°2 | 62°0 |.60°2 | -522 | 87 5 
6 20 5, |29°93 | 64°0 | 62°0 | 6073 | 524 | 88 8 | ... |Cirrocumulus ; sun shining on 
dome of Great Equatorial. 
6 30 5, | 29°93 | 63°7 | 62°0 | 60°6 |*529 | 90 | s.w. | 10 | «. |Overcast. : 
6 40 5, | 29°93 | 63°4 | 61°9 | 60°7 |°531 | gt 10 eee. cirrocumulus, cu- 
6 50 5, | 29°93 | 63°3 | 61°7 | 60°4 | °526 | go se EO} [ler mulus, and cirrostratus. 
62°7 | 61-5 | 60°5 |*528 | 94 | s.s.w.| 10 | ... |Overcast. 
| 
Report on the Theory of Numbers—Part IV. By H. J. Stepney 
Smita, M.A., F.R.S., Savilian Professor of Geometry in the Univer- 
sity of Oxford. 
105. General Theorems relating to Composition.—The theory of the compo- 
sition of quadratic forms occupies an important place in the second part of 
the 5th section of the ‘ Disquisitiones Arithmetice,’ and is the foundation of 
nearly all the investigations which follow it in that section. In accordance 
with the plan which we have followed in this portion of our Report, we shall 
now briefly resume the theory as it appears in the ‘ Disquisitiones Arithmeticee,’ 
directing our special attention to the additions which it has received from 
subsequent mathematicians. We premise a few general remarks on the 
Problem of composition. 
If F, («,, v,,...,) be a form of order m, containing n indeterminates, 
which, by a bipartite linear transformation of the type 
U,=2 My, B,y Vp %y, 
21,232,853 0, 
P=1,,2,3,...2, [7 
-y=l1, 2, 3, bh te 
