TRACT 37.

OF GUNNERY^

S09

quantity = 0, the equation reduced gives hyp. log. of

— — — 6d + a — l-l-—, a general expression for the value

a 10 -6d T 106d’ 5 1

of the hyp. log. of —, in assuming the whole value of p, or

half the weight of the powder in the formula.

185. Now, though it might be very easy to find the valueof a in this equation, for any particular values of b and d, bymeans of the rule of double position; yet it may be properto obtain an equation, and a general value of the length ofthe charge a, out of logarithms. And for this purpose it maybe convenient to employ Dr. Halley’s approximation for anumber from a given logarithm, which is this, the number is~ = g-i-j, where — is the required number, and l its hyper-bolic logarithm. Now, in the present case the quantity ldenotes the fraction 10 6d + a - which therefore being substi-

10'6a 3

tutcd for l, that number becomes — = ———^ = 3 + —.

Hence it appears that, in all cases, the value of a is less than■j- of b, and that so much the more as the charge is higher.The diameter d being = l - 96, which substituted for it, thelast equation becomes — = 63 - + - : which reduces to this

* a 21 — a

quadratic equation, a z + (65 -)- b) a — 21 b. This appliedto our first gun, where b = 28 - 53, it gives a = 6f; hence= 4'6 nearly ; which by the experiment was —

—But when the same quadratic equation is applied to the4th gun, in which b = S0^, it brings out a = 11 nearly;consequently nearly, but which by the experiment

was 2 T 7 = 6|. So that, in every case, the theorem brings outthe value of a rather too great, when the whole value of p ,or the half weight of the charge, is used in the denominatorp + w.

186. But, we before found, in determining the velocityof the inflamed elastic fluid, that the properest value of thequantity p , is not equal to half the weight of the charge, butrather to £ of it, or to f p. By using this quantity then, in-’