  Two Latin squares $L=[l(i,j)]$ and $M=[m(i,j)]$, of even order $n$ with
entries $\{0,1,2,\ldots,n-1\}$, are said to be nearly orthogonal if the
superimposition of $L$ on $M$ yields an $n\times n$ array $A=[(l(i,j),m(i,j))]$
in which each ordered pair $(x,y)$, $0\leq x,y\leq n-1$ and $x\neq y$, occurs
at least once and the ordered pair $(x,x+n/2)$ occurs exactly twice. In this
paper, we present direct constructions for the existence of general families of
three cyclic mutually orthogonal Latin squares of orders $48k+14$, $48k+22$,
$48k+38$ and $48k+46$. The techniques employed are based on the principle of
Methods of Differences and so we also establish infinite classes of
"quasi-difference" sets for these orders.
