By A. R. Forsyth
Vintage 19th-century paintings one in all the best remedies of the subject. Differential equations of the 1st order, common linear equations with consistent coefficients, integration in sequence, hypergeometric sequence, resolution by means of convinced integrals, many different themes. Over 800 examples. Index.
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Extra resources for A Treatise on Differential Equations
By Theorem 2 there exist at least two lines. If they are to pass through P and be distinct, they must contain points not in common, so let us say that Zcontains P and Q and m contains P and R. By Axiom la there exists a line, say k, containing Q and R ; by assumption, it must contain P. 4 Further Proofs in Axiomatic System 3 31 this contradicts Theorem 4. Hence, our assumption that every line passes through P must be false. Hence, there exists at least one line not through P. • Theore m 6. There exist at least three lines through any point.
We shall consider this a direct proof. The only proof which we will call indirect is one which begins immediately by contradicting (or denying and considering all cases) the statement we are attempting to prove. Let us analyze the proof step by step. STEP 1 : Axiom 2 together with Axiom 4 is an instance of argument form (a) . STEP 2 : Axiom 3 together with Axiom 4 is another instance of pattern (a) . STEP 3: Axioms l a and 1 b together with the results of Steps 1 and 2 are another instance of pattern (a) .
4. There exists at least one line. 5a. If Z is a line and P a point not on it, then there exists at least one line m through P with no point in common with Z. 5b. If Z is a line and P a point not on it, then there exists at most one line m through P with no point in common with l. Axio m Set 2 la. If P and Q are any two points, then there exists at least one line containing both P and Q. 1 b. If P and Q are any two points, then there exists at most one line containing both P and Q. 2. If Z is a line, then there exist at least three points on it.