THE KOYAL ARTILLEKY INSTITUTION. 
189 
Again , 
__ctv _ g$_ 
/= 
dt 
(A + fxv~), 
Art. 86, 
- 7 = • d(v</n) 
w 
A -+• v*/(j\ 
... 1 . * (tan-i — ^ -tan-i^A) = -''fU; hr«=Fv}hent=0....(l) 
Vp A a/A. > » 
Equations (y?) and (^) give tlie length and time of penetration whilst 
the velocity is reduced from F to v. 
a Vis viva ” is a term used to express the mass of a body multiplied Art. 87. 
by the square of its velocity. When the velocity is reduced from V to v, 
w 
the loss of vis viva is M (V° — v 3 ); where M = mass = -. 
. 9 
The “ work done 33 is half the loss of vis viva. 
The ordinary expression for volume of revolution is °/ x 7rg 2 dx 9 but 
here the notation " s 33 is used in exactly the same sense as “ x. 33 
Hence the equation 
l - {V* - v°~) oc °/y& = k yy&; 
where ~k is some constant depending on the nature of the earth and on 
the form of the projectile. 
Conclusion. 
The object of Professor Bashforth was to calculate the motion of Chap. II. 
projectiles in air, and the chapter on motion in vacuo was only intro¬ 
ductory. These notes, therefore, have only referred to the symbols 
“ 2b 33 “ X, 33 “ Y 33 “ T/ 3 with reference to the articles on motion in 
air; for any previous explanation of them would have had to be 
repeated. If, however, the writer has succeeded in removing diffi¬ 
culties, the symbols a X Q e , &c., in Art. 35 will be perfectly intelligible; 
of course y = 0, since there is no resistance. These symbols are 
merely used in this article for the sake of symmetry with Art. 63, § 11. 
although the integrals they express are easily solved. 
With regard to “2h 33 this symbol is first used in Art. 44; conse¬ 
quently this passage may cause difficulties to those who are reading 
the book for the first time. The whole investigation of the law of 
resistance in Chap. III. must be first studied, and then it will be seen 
that 2 bP is the same as A % since 2b has been put for the numerical Art. 56. 
A H § 9. 
value of -y 2 -. Hence the fourth column of figures in Art. 44 is merely 
a table of half the second differences of the times. The last column is 
a taWe of TUP > since K ~ I ^ |S 2& Js* 
Equation (V.) in Appendix III. is worth studying with regard to its 
