160 Transactions of the Royal Society of South Africa. 
and, with respect to y, 
There thus follows on substituting the results of the preceding paragraph 
8. Witli the same notation as before 
'dx ~ ' / i^x ~dy ~ ' X i ^y ^y ~ ' 
jdy '^x 
The first of these is readily seen to be obtainable by differentiating with 
respect to x the first result of § 6, and then applying the first result of § 4. 
The others are reached in quite similar fashion. 
9. With tlie same notation as before 
ox^ / I ^oxoy Z_i '07/2 
Differentiating the result of § 3 twice with respect to x we obtain 
similarly one differentiation with respect to x followed by one with respect 
to y gives 
/ / ^'dx'dy /Lj'dy 'dx / i^y ~dx / / ^~dx'dy~ ' 
and two differentiations with respect to y give 
^^y' ^ / i ^y ^y z i ^^y~ 
And as in each of the three expressions here all the terms except the first 
have been shown to vanish, it follows that the first terms also must 
vanish. 
