Dec. 14, 1857.] GALTON ON THE EXPLORATION OF ARID COUNTRIES. 65 
is drawn up under the several conditions of one, two, or three meals 
being issued per day, and the details of any particular set of cases 
can, at once, be deduced from it. 
The “ differences ” supply ready means for continuing the columns, 
and, if it be desired to extend the table, or to work out any 
question relating to the subject of it, it is easy to do so by em- 
ploying the following formulae. Let a - the number of the explor- 
ing party ; b ~ those of the section that turns back last ; s = those 
of the section that turns back first ; m = total number of individuals 
(horses or men, as the case may be) employed in the expedition ; 
l - load of day rations carried by each of them ; n — number of 
meals into which the day ration is divided ; d = distance to which the 
explorers are to be forwarded ; r = number of stages into which d is 
divided; g = length of each of them reckoned in day’s journey. 
Then, we have — 
( 1 .) 
( 2 .) 
(3.) 
( 4 )- 
a 
b = 
m — 
In — • 2 gn + 1 . 
2 gn. 
O+fc) r 
a M 
(a -f- 0 r 
(a + b)-' ~ 
d 
O + ZQ '- 1 
"(a + b ) r - ! 
(5.) The additional distance that can be travelled by a , after the 
last section has turned back, = ^ ; if this is not integral, the 
A n 
integer next less than it must be adopted. 
In the equation (2), if a - 1, and if b is an integer, then m 
is necessarily an integer also ; but if a be greater than 1 , and if 
be a fraction reduced to its lowest denomination, m cannot be 
integral, unless both it and all the terms of the series a, b, , . . s, 
be multiplied by a T ~\ This has been done in the headings of the 
fourth and subsequent columns of Table 4, and shows the cumbrous 
a . 
series of terms that become necessary when - is other than a very 
simple fraction. An accurate adjustment of loads taken and of food 
wanted is not, however, essential, and I have therefore appended an 
approximative series of terms of more manageable size, and 
which would serve well enough, in actual practice, in cases 
where each man’s rations were not kept separate from those of his 
colleagues. 
Table 5 : it shows the precise method according to which the 
