298 Professor Hamitton on Conjugate Functions, 
CONTENTS OF THE PRELIMINARY AND ELEMENTARY ESSAY ON ALGEBRA 
AS THE SCIENCE OF PURE TIME. 
Articles, 
Comparison of any two moments with respect to identity or diversity, subsequence or precedence 
Comparison of two pairs of moments, with respect to their analogy or non-analogy.......-..-+--+-+-+- Hea 3. 
Combinations of two different analogies, or non-analogies, of pairs of moments, with each other.........sssseeeeeeeeee 4 
On continued analogies, or equidistant series Of MOMENES... ...cesssceseeeecsseeeeesctseeeeteee teen setae PEO ort: ced 5,6,7,3 
On steps in the progression of time: their application direct or inverse to moments, so as to generate other moments ; 
and their combination with other steps, in the way of composition or decomposition........---.++sees++eeuwee sree0 9,10,11,12 
On the multiples of a given base, or unit step ; and on the algebraic addition, subtraction, multiplication, and division, 
of their determining or multipling whole numbers, whether positive or contra-positive, or null.. ............:24 13,14,15 
On the submultiples and fractions of any given step in the progression of time ; on the algebraic addition, subtraction, 
multiplication, and division, of reciprocal and fractional numbers, positive and contra-positive ; and on the impos- 
sible or indeterminate act of submultipling or dividing by ZeT0...--...sscseseeeeneeseseceeeesneeeeeeesteresaneres -16,17,18,19,20 
On the comparison of any one effective step with any other, in the way of ratio, and the generation of any one such 
step from any other, in the way of multiplication; and on the addition, subtraction, multiplication, and division of 
algebraic numbers in general, considered thus as ratios or as multipliers of stepS......---sss01 creseserees: ceeeeeee/D1,22,23 
On the insertion of a mean proportional between two steps ; and on impossible, ambiguous, and incommensurable 
square roots Of ratiOS...........scee---eeeee seers paodccosomonoscoabessaeoser wsoanesn fone. do tsans==n-ne cecscsrndoncereass PS -caccciod 24,25 
More formal proof of the general existence of a determined positive square-root, commensurable or incommensurable, 
for every determined positive ratio; continuity of eee of the square, and principles connected with this 
continuity.......--.+ Ss rete Waters stee iecavarcsansreaeecesopeesenscenenssacseasene-suasaneacserssenessrauccc! ube cubinwsinmnsbieigsas «40g= apie 26,27 
On continued analogies or series of siaconlinial steps ; va on powers and roots and logarithms of ratios...28,29,30,31 32,33 
Remarks on the notation of this Essay and on some modifications by which it may be made more like the notation 
commonly employed ........... oc CerECEe Sioteateait cence oes aavaaeaues Unahe than cons sense pmactunentaarsthawnetpasanssmesenas er tod: + 34,29,36 
CONTENTS OF THE THEORY OF CONJUGATE FUNCTIONS, OR ALGEBRAIC COUPLES. 
= Articles. 
On couples of moments, and of steps, in time..........--.seesecseeescssseeeeseteetnes cocrenersersesserrsesassnaseesenannanearscansnes 
On the composition and decomposition of step-couples......... 
On the multiplication of a step-couple by a number 
On the multiplication of a step-couple by a number-couple; and on the ratio of one step-couple to another............-+ 
On the addition, subtraction, multiplication, and division, of number-couples, as combined with each other.. 
On the powering of a number-couple by a single whole number............ sdinevuetidaunaceseqashtccentssshsu=-saHsssa5 0 aeigeeeMEan 
On a particular class of exponential and logarithmic function-couples, connected with a particular series of integer 
chencAaasreccoeeoos pope Ebo coe co 2800-5 noses cheba Meera rocaanaraceca ccoscocopeacadsoecobo cL MIS 
On the powering of any number-couple tea any single number or number-couple 
powers of number-couples. 
On exponential and logarithmic function-couples in general. 
