98 
MR. E. P. CULYERWELL ON DISCRIMINATION OF MAXIMA AND 
articles. Those who desire to read only the general method will find Part II. 
complete in itself. 
It was originally intended that a tolerably complete accoimt of the treatment of 
the problem, when the limits were not all fixed, should be inserted, but the length to 
which the paper has extended seems to render this inexpedient. 
Part I. 
Algebraic Transformations of the Second Variation. 
1. Notation. 
2, 3, and 4. General remarks on the problem for two variations. “ Synclastic” and 
“ anticlastic ” functions. 
5 and 6. Examination of Jacobi’s method. 
7 and 8. Comparison with algebraic method of this paper. 
9 and 10. Two variables—general case. 
11. Probable failure of the transformation if the limits widely separated. 
12. Criterion given by the result of the transformation. 
Part II. 
The General Method. 
13 and 14. Conditions implied in the problem. 
15. S 2 V and dfjdif have the same sign for small range of integration. 
16. Integration—limits within which the property holds. 
17. General remarks on the foregoing proof. 
18. Any number of variables—notation and limitations. 
19. Statement of the general problem. 
20. Criterion for the sign of S 3 U where the “ highest fluxions of any dependent 
variable are all of the same order, the integration extending over a small 
region only. 
21. Limits within which the criterion is sufficient. 
22. The “ highest fluxions ” are not all of the same order. The result includes 
that of Art. 20. 
23. The highest of all the fluxions of any dependent variable appears in the first 
degree only. 
Part I. 
Algebraic Transformations of the Second Variation. 
1. In the following pages the word “fluxion” will be used instead of the long 
expression “ differential coefficient.” When there is but one independent variable, 
x, the successive fluxions of the dependent variables will be expressed in the known 
notation 
